Let $Alg$ be a single-sorted finitary variety i.e. one specified by a set $\Sigma$ of operations with finite arity and also a set $E$ of equations, the morphisms being the usual $\Sigma$-algebra morphisms.
Further let $T : Alg \to Alg$ be a finitary functor that preserves monos (injections) and empty binary intersections i.e. $T(A \cap B) = TA \cap TB$ whenever $A \cap B = \emptyset$. This trivially holds if $\Sigma$ contains a constant.
Then my question is:
Does there exist such a $T$ that fails to preserve finite or infinite intersections?
Note that every $T : Set \to Set$ preserves non-empty finite intersections by page 3 of:
so under my assumptions $T$ would preserve finite intersections. Then if $T$ is finitary this can be shown to imply preservation of infinite intersections. Therefore no such example can be found in $Set$, this being the variety where $\Sigma = E = \emptyset$.
Any help much appreciated.